The use in the past of mathematical techniques and methodologies to design macro-world structures in the past are now considered for designing micro-world structures. In this work, a computational algorithm for estimating natural frequencies and mode shapes of cantilever microbeams is presented. The methodology is based on the Rayleigh-Ritz method, which considers the potential and kinetic energies of the mechanical system. A proposed polynomial equation is first considered to establish the behavior of the microstructure. Together with the energy equations, an eigenvalue problem is constructed. The corresponding eigenvalues give an approximate estimation of the frequency values; the corresponding eigenvectors once substituted in the proposed polynomial describe the mode shapes of the microbeam for each frequency value. The results obtained with this methodology were compared with the finite element method. An interesting difference between the methodologies and the algorithm was found and the corresponding results are shown.